refactor: extract status manager

[garritfra]

Summary

• Add StatusManager to own App status messages and expose set/clear/read helpers.
• Route dispatch, search, render, and event-loop status access through the manager.
• Complete the final explicit extraction slice from Decompose app.rs god object into focused subsystems #73 after SearchState and CommandState.

Closes #73.

Test plan

• cargo fmt --all
• cargo clippy --workspace --all-targets --all-features
• cargo test

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